Luc Florack - Böcker

This text covers Gaussian scale-space theory from its applications to its mathematical foundation. The reader not so familiar with scale-space should find it instructive to first consider some potential applications described in Part I. The next two parts both address fundamental aspects of scale-space. Whereas scale is treated as an essentially arbitrary constant in Part II, Part III emphasises the deep structure, i.e. the structure that is revealed by varying scale. Finally Part IV is devoted to non-linear extensions, notably non-linear diffusion techniques and morphological scale-spaces, and their relation to the linear case. The book is addressed to researchers in the field of image analysis seeking mathematical foundation of algorithms.

Despite the fact that images constitute the main objects in computer vision and image analysis, there is little concern about their actual definition. In this book, a complete account of image structure is proposed in terms of rigorously defined machine concepts, using basic tools from algebra, analysis, and differential geometry. Machine technicalities such as discretisation and quantisation details are de-emphasized, and robustness with respect to noise is manifest.

Mathematical Methods for Signal and Image Analysis and Representation presents the mathematical methodology for generic image analysis tasks. In the context of this book an image may be any m-dimensional empirical signal living on an n-dimensional smooth manifold (typically, but not necessarily, a subset of spacetime). The existing literature on image methodology is rather scattered and often limited to either a deterministic or a statistical point of view. In contrast, this book brings together these seemingly different points of view in order to stress their conceptual relations and formal analogies.Furthermore, it does not focus on specific applications, although some are detailed for the sake of illustration, but on the methodological frameworks on which such applications are built, making it an ideal companion for those seeking a rigorous methodological basis for specific algorithms as well as for those interested in the fundamental methodology per se.Covering many topics at the forefront of current research, including anisotropic diffusion filtering of tensor fields, this book will be of particular interest to graduate and postgraduate students and researchers in the fields of computer vision, medical imaging and visual perception.

Mathematical Methods for Signal and Image Analysis and Representation presents the mathematical methodology for generic image analysis tasks. In the context of this book an image may be any m-dimensional empirical signal living on an n-dimensional smooth manifold (typically, but not necessarily, a subset of spacetime). The existing literature on image methodology is rather scattered and often limited to either a deterministic or a statistical point of view. In contrast, this book brings together these seemingly different points of view in order to stress their conceptual relations and formal analogies.Furthermore, it does not focus on specific applications, although some are detailed for the sake of illustration, but on the methodological frameworks on which such applications are built, making it an ideal companion for those seeking a rigorous methodological basis for specific algorithms as well as for those interested in the fundamental methodology per se.Covering many topics at the forefront of current research, including anisotropic diffusion filtering of tensor fields, this book will be of particular interest to graduate and postgraduate students and researchers in the fields of computer vision, medical imaging and visual perception.

Whatisactuallytheinformationdirectlyrepresentedinthescale-space?Istarted to wonder about this shortly after Peter Johansen, 15 years ago, showed me his intriguing paper on how uniquely to reconstruct a band-limited 1D signal from its scale-space toppoints. Still, I have not fully understood its implications. Merely recording where structure vanishes under blurring is su?cient to fully reconstruct the details. Of course, technicalities exist, for example, you must also know negative scale toppoints. Nevertheless, I ?nd it surprising that we may trade the metric properties of a signal with the positions of its inherent structure. The result has been generalizedto analytic signals, shown also for the zero crossings of the Laplacean, but has not yet been generalized to 2D. This remains an open problem.In 2003, Peter Giblin, Liverpool University, Luc Florack, Eindhoven Univ- sity of Technology, Jon Sporring, University of Copenhagen, my colleague Ole Fogh Olsen, and several others started the project collaborationDeep Structure and Singularities in Computer Vision under the European Union, IST, Future and Emerging Technologies program, trying to obtain further knowledge about what informationis actuallycarriedby the singularitiesof shapesand gray-scale images. In this project, we probed from several directions the question of how much of the metric information is actually encoded in the structure of shapes and images. We, and many others, have given hints in this direction.

This book constitutes the refereed proceedings of the First International Conference on Scale-Space Theory for Computer Vision, Scale-Space '97, held in Utrecht, The Netherlands, in July 1997.The volume presents 21 revised full papers selected from a total of 41 submissions. Also included are 2 invited papers and 13 poster presentations. This book is the first comprehensive documentation of the application of Scale-Space techniques in computer vision and, in the broader context, in image processing and pattern recognition.

This book covers Gaussian scale-space theory from its applications to its mathematical foundation. The reader not so familiar with scale-space will find it instructive to first consider some potential applications described in Part I. The next two parts both address fundamental aspects of scale-space. Whereas scale is treated as an essentially arbitrary constant in Part II, Part III emphasises the deep structure, i.e. the structure that is revealed by varying scale. Finally Part IV is devoted to non-linear extensions, notably non-linear diffusion techniques and morphological scale-spaces, and their relation to the linear case. Audience: This volume is addressed to researchers in the field of image analysis seeking mathematical foundation of algorithms.